A Stable and Efficient Algorithm for the Indefinite Linear Least-Squares Problem

Abstract

We develop an algorithm for the solution of indefinite least-squares problems. Such problems arise in robust estimation, filtering, and control, and numerically stable solutions have been lacking. The algorithm developed herein involves the QR factorization of the coefficient matrix and is provably numerically stable.

A backward stable hyperbolic QR factorization method for solving indefinite least squares problem

“…Finally computing x needs about 2n 2 flops. Table 1 compares the cost of Algorithm 1 with the costs of the methods proposed in [5] and [2]. So about the cost Algorithm 1 is between other two methods.…”

“…Algorithm in [5] Algorithm in [2] (4p + 4q − 4n/3)n 2 or (2p + 4q + 2n)n 2 (5p + 5q − n)n 2 (2p + 2q − 2n/3)n 2 p >> n (2p + 4q)n 2 (5p + 5q)n 2 (2p + 2q)n 2 p ≈ n (8n/3 + 4q)n 2 (4n + 5q)n 2 (4n/3 + 2q)n 2 …”

“…By pre-multiplying R T , it is just the normal equation (3). In [5] it is shown that this method is numerically backward stable. The method proposed in [2] uses the hyperbolic QR factorization, an analog of the QR factorization, to solve the ILS problem.…”

“…This problem has several applications. Examples include the total least squares problems ( [5]) and the H ∞ smoothing problems ( [7,10]). It is known that the ILS problem has a unique solution if and only if…”

“…Following the idea of the QR factorization method, recently two methods were developed to solve the general ILS problem. The method proposed in [5] uses the QR factorization of A to solve the ILS problem. The precise procedure is as follows.…”